Monthly Archives: August 2011

This weeks sandwich request comes from the inimitable Paul ‘Mandragoran’ Quigley (just back from Switzerland as part of the Irish team in the 40k world championships!). We’re taking another look at penetration today, this time factoring in the impact of rending. A useful reference point would be here where I’ve covered the efficacy of a variety of anti tank weapons, but the material in this post does stand on its own too. Right, question time:

A Black Legion tank commander is navigating his Land Raider through enemy territory. He’s caught in crossfire between two Razorbacks, one left, and one right. Both are at the limit of their weapon range, so the commander decides to rush towards one so that he can’t be hit by both. Should he drive toward the Assault Cannon armed Razorback, or the Lascannon armed Razorback?

So, rending against vehicles. You get your usual 1d6 plus weapon strength roll, with the added twist that if you get a 6 on the roll, then you get to roll a d3 and add it to your result. This means that an assault cannon at Str 6 could get a 15 and penetrate a Land Raider (Str6+roll a 6 on d6+ roll a 3 on d3 =15). The question is, can it do a better job than a Lascannon (which is a plain Str 9 + 1d6 with no rending)?

It’s a little tricky to do a straight like-for-like comparison here as a key feature of the assault cannon is that it gets four shots, whereas the Lascannon only gets one. So here’s my starting point. I’m assuming that there are no misses, and for the Lascannon I’m looking at the odds of a penetrating hit, and for the assault cannon I’m looking at the odds of at least one penetrating hit. Some of you may object to that approach, but bear with me for now.

(Just to note: the gap in the line isn’t a mistake, it’s simply that you can’t get a 12 on the assault cannon as rolling the 6 gets you an extra d3 making you jump from 11 to 13+).

So what have we got: one hit with a lascannon has a (just under) 17% chance of getting a pen on AV14, but 4 hits from an assault cannon gets you (just over) 17% chance of getting one or more penetrating hits on AV14. Okay the odds are only a tiny bit higher but you can get from one to four pens so the net effect can be a lot stronger. So the word on the street is correct, assault cannons are straight up better than lascannons at penetrating AV14?

Well, not so fast.

Let’s go back to those assumptions from earlier. None of the shots miss. “So what?” you say, “the assumption was the same for both!“. Actually it’s different. The odds of getting all hits on a one shot weapon are better than the odds of getting all hits on a four shot weapon (assuming equal BS). The analysis above assumes that the Razorbacks never miss, so the more unreliable the firer, the less accurate that graph becomes.

To illustrate the point I’ve run the same analysis showing the results for 4, 3, 2 and 1 hits on the assault cannon versus the lascannon.

The comparison is no longer quite so clear cut. We need to account for the end to end process from hitting through to penetrating. So lets’ do that. Lets assume BS4 for the assault cannon and the lascannon.

For the assault cannon we need:

3+ to hit

6+ to rend

5+ to penetrate

This gives us a 4% probability of success. But we get 4 shots, if you’re thinking that 4 shots at 4% gets you a 16% probability of success then you probably need to read my blog more often; if you’re thinking the answer is 14% then you probably don’t need me at all. (Success here means one or more penetrate results)

For the Lascannon we need:

3+ to hit

6+ to penetrate

This gives us an 11% probability of success, and since we only get one shot, that’s the total odds.

So what’s the final verdict? The analysis clearly shows that neither the assault cannn or lascannon are particularly good at killing Land Raiders, but if they both have the same BS, the assault cannon is definitively better. It is worth noting that the lascannon can sneak ahead if fired by a superior marksman, so a BS5 lascannon is equal to a BS4 assault cannon, and a twin linked BS4 lascannon is better than a BS4 assault cannon (vs AV14).

I say final verdict, but there’s still a little more gas in the tank. I’ve plotted a couple of different weapons so you can check out the relative merits of weapons that I haven’t covered previously. Note that I’ve not done the full end to end calculation here, I’ve simply assumed all shots hit for this chart.

One final weirdness I wasn’t quite expecting, against the humble rhino (AV11) the lascannon is more reliable. It’s basically an artifact of the rend: if you get a 6 then your result ‘jumps up’ out of line with the non rending results. So while we initially were concerned only with AV14, we can in fact make a more general statement: AV12 and above, assault cannon more lethal, AV11 and below Lascannon more deadly!

I first touched on scatter dice in my earlier post on deep striking. Blast weapons don’t use Ballistic Skill in the same way as ‘regular’ weapons, but it is still an important factor in hitting your foes. Question time:

A renegade Ordo Hereticus Inquisitor carrying a psyocculum is hunting the battlefield for Mephiston. The inquisitor is joined by his trusty squad of psyker henchmen with the Psychic Barrage (large blast) power. Assuming they pass their psychic test, what are the odds that they will hit Mephiston?

Right, blast weapons use the scatter dice described here. At its most basic, a 33% chance of a hit, and 67% chance of a miss; if you miss then it scatters 2d6 inches but unlike deep striking you can subtract the firing models Ballistic Skill (BS) from the 2d6 result. So if you roll a ‘miss’ but get a distance less than or equal to your BS then that miss becomes a hit (i.e. you don’t scatter). Naturally this means that the higher your BS, the more ‘misses’ get converted into hits, and if it does scatter then it won’t scatter as far.

To show the effect of increasing BS values I’ve pulled together a 3d plot. So each colour represents a BS value, from BS0 at the front to BS10 at the back. The odds of a particular result go from left to right, so taking BS0 as an example, the odds of a HIT is the leftmost blue column (at 33.33%) and the odds of a particular scatter are to the right, e.g. a 2 inch scatter has a probability of 1.85% (for BS0) and a 7 inch scatter would be 11.11% likely (for BS0).

So quick summary on how to read this:

the height of a given column is the probability,

each colour is a BS value, and the BS values get higher as you go back,

the foreground numbers are a HIT (leftmost) or a particular scatter distance (from 1 to 12 inches).

Since the the BS value is subtracted from the scatter distance, you can clearly see the maximum scatter get smaller with each step increase in BS. So looking at the BS10 scatter (all the way at the back in pink) it’s a 94% chance of a hit, 4% chance of a 1 inch scatter, and a 2% chance of a 2 inch scatter.

Given that the the psyocculum gives our Psykers BS10, is that the answer to the question, a 94% chance of hitting Mephiston?

Not quite, theres one more factor to take into consideration. Blast size. The regular blast has a 1.5 inch radius, and the large blast has a 2.5 inch radius. Against a vehicle, only the centrepoint of the blast gets you a full strength hit, but against infantry just clipping the base with the blast template is enough for the full whack.

So in terms of hitting Mephiston, it’s 2d6 scatter minus 10 for BS, and (effectively) minus another 2.5 inches for the radius of the large blast. So we’re subtracting 12.5 from a number that is at most 12, simply put they can’t miss! There aren’t many mechanics in the game that can say that.

The only caveat is that the extra bit of reach from size of the radius doesn’t get you a ‘full’ hit as it doesn’t land exactly where you placed it. You’ll definitely hit the guy you were centred on, but you’ll cover different models around him if it does scatter those one or two inches.

So, back to more general principles. You may recall my uber list of BS rankings, well I’ve now we can add two new charts to that list.

First up, regular BS accuracies with blast accuracies added (in pink). The blast accuracies are the probability of hit but disregarding the radius of the blast (i.e. the odds of the centrepoint hitting your desired target point). Blue columns are ‘regular’ shots, green are twin linked, and pink are blast. Hmmm I guess I left out twin linked blast, guess that’ll have to wait.

Secondly looking specifically at the effect of the radius (no ‘normal’ i.e. non-blast shots on this one) . So these are the odds of hitting with just the centre (in pink), versus blast (in blue), versus large blast (in green).

The pattern is actually pretty simple: BS10 centre is as accurate as BS9 blast, is as accurate as BS8 large blast (and so on down). Essentially each step up in blast size is equivalent to a one point increase in BS.

So there you have it – I thought I had all the BS covered, but there was still more to do; …always more to do.

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As I mentioned yesterday, Warhammer 40,000 doesn’t quite have a world championship, but this role is at least partially fulfilled by the growth of the European Team Championship (ETC) to incorporate teams from all over the world, far beyond its European origin.

While it is a highly competitive international event, it’s important to retain some perspective, and ensure that all players enjoy the complete range of experiences on offer, as clearly demonstrated by Woody:

ETC 2011 finished in Switzerland today with the final three rounds of the two-day six-round tournament. What I’ll cover here is a brief round up of the results, mainly focusing on the performance of team Republic of Ireland, and team Northern Ireland.

Team NI faced off Russia for their opening game, but the great Bear of Russia proved a difficult opponent. Loss for Team NI. Republic of Ireland looked like a victory mid game, but the Danes pulled back the draw. Not a bad start for RoI, top half of the table!

Worth noting that this was the first ever loss in a round by Poland, so well done to the German team for that performance. I’m sure there’s an inappropriate WW2 joke in there somewhere, but let’s just leave it at that.

ROUND 5

Team A

Team B

Points

TP

Sweden

Germany

2 : 0

99 : 61

Poland

Spain

2 : 0

95 : 65

France

United States

0 : 2

61 : 99

Italy

England

0 : 2

74 : 86

Latvia

European Union

2 : 0

112 : 48

Ireland

Switzerland

0 : 2

55 : 105

Wales

Czech Republic

2 : 0

86 : 74

Austria

Belarus

0 : 2

74 : 86

Denmark

Scotland

1 : 1

75 : 85

Finland

Belgium

2 : 0

119 : 41

Norway

Russia

0 : 2

50 : 110

Northern Ireland

Greece

1 : 1

76 : 84

Team RoI faces the host nation, Switzerland, and lets them win out of sheer politeness. Team NI get back on the horse with a hard fought draw against Greece. Things are heating up at the top, with Germany, Poland and Sweden all on 8 points.

ROUND 6

Team A

Team B

Points

TP

Poland

Sweden

2 : 0

103 : 57

Latvia

Germany

0 : 2

47 : 113

England

United States

0 : 2

67 : 93

Switzerland

Spain

0 : 2

66 : 94

Wales

Belarus

2 : 0

100 : 60

Italy

France

0 : 2

71 : 89

Finland

European Union

2 : 0

90 : 70

Denmark

Czech Republic

2 : 0

91 : 69

Russia

Scotland

1 : 1

83 : 77

Ireland

Austria

2 : 0

103 : 57

Belgium

Greece

2 : 0

91 : 69

Norway

Northern Ireland

- : 160

RoI took on Austria in the final round, but flight schedules meant the Austrians couldn’t stay the course and had to concede. Not the way RoI would like to win it, but there you go. Similarly Team NI got a walkover from Norway as they had to leave early. A bit of a flat end, but the boys did us proud and hopefully they had a great time!

FINAL RESULTS

Place

Team

Points

Difference

1

Germany

10

258

2

Poland

10

174

3

United States

9

160

4

Sweden

8

158

5

Spain

8

58

6

France

8

52

7

Wales

8

-6

8

England

7

36

9

Finland

7

36

10

Switzerland

7

24

11

Latvia

7

22

12

Denmark

6

-6

13

Ireland

6

-56

14

Belarus

5

36

15

Italy

5

24

16

Scotland

5

-6

17

Russia

5

-26

18

European Union

5

-48

19

Belgium

5

-120

20

Northern Ireland

4

-40

21

Czech Republic

3

-10

22

Austria

3

-40

23

Norway

2

-416

24

Greece

1

-264

Great performances overall from teams RoI and NI, and we hope for even better next year! For excellent video coverage of the full event, check out RHQ.tv

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Warhammer 40,000 doesn’t quite have a world championship, but this role is at least partially fulfilled by the growth of the European Team Championship (ETC) to incorporate teams from all over the world, far beyond its European origin.

ETC 2011 takes place in Switzerland today (Saturday 20th August) and tomorrow (21st). I’ll be doing a brief round up at end of both days, mainly focusing on the performance of team Republic of Ireland, and team Northern Ireland.

A good start for team Ireland, beating Finland in a close match up. The first game of a tournament can really make or break players’ confidence and this got the whole team on a good footing. The Northern Ireland team held their own against Austria, all to play for in round 2!

ROUND 2

Team A

Team B

Points

TP

Germany

United States

2 : 0

105 : 55

Spain

Scotland

2 : 0

97 : 63

Latvia

Poland

0 : 2

57 : 103

Ireland

Belgium

1 : 1

82 : 78

Switzerland

France

0 : 2

59 : 101

Italy

Austria

2 : 0

86 : 74

Sweden

Northern Ireland

2 : 0

121 : 39

England

Belarus

1 : 1

79 : 81

Finland

Russia

2 : 0

88 : 72

Norway

Denmark

0 : 2

44 : 116

Czech Republic

European Union

1 : 1

83 : 77

Wales

Greece

2 : 0

108 : 52

A good start in Round 1 means a tough opponent in Round 2! The RoI team faced off Belgium, and came away with a draw. Another solid performance which puts them in a good position for Round 3.

Joe “Maynard” Cullen wore his lucky hat and beard, and it appeared to pay off getting 11-9 with his Chaos Marines against seer council Eldar:

Team Northern Ireland lost their way slightly in round 2 and were defeated by the Swedish team.

As a twist of fate, RoI would face Sweden next in Round 3, can they get revenge for their Northern brethren?

ROUND 3

Team A

Team B

Points

TP

Germany

Spain

2 : 0

107 : 53

Poland

France

2 : 0

112 : 48

Ireland

Sweden

0 : 2

38 : 122

Italy

Belgium

2 : 0

106 : 54

Denmark

United States

0 : 2

55 : 105

Latvia

Scotland

2 : 0

97 : 63

Finland

England

0 : 2

55 : 105

Wales

Austria

2 : 0

97 : 53

Switzerland

Belarus

2 : 0

92 : 68

Northern Ireland

European Union

0 : 2

51 : 109

Russia

Czech Republic

0 : 2

32 : 128

Norway

Greece

2 : 0

92 : 68

After a poor performance in Round 1, the Swedes were fighting hard to get back up to the top. Team Ireland stood in their way at the end of day 1 but unfortunately got steamrolled by the Nordic Nightmare. A harsh result for RoI, but tomorrow will be a chance to get back in the tournament. Team NI suffered at the hands of the Merc term, they’ll need a good start tomorrow to lift their spirits!

OVERALL PLACING END OF DAY 1

Place

Team

Points

Difference

1

Germany

6

184

2

Poland

6

144

3

Sweden

5

162

4

Italy

5

68

5

United States

4

100

6

Spain

4

46

7

England

4

40

8

Latvia

4

34

9

Wales

4

20

10

France

4

-6

11

Czech Republic

3

36

12

European Union

3

0

13

Switzerland

3

-10

14

Belgium

3

-28

15

Ireland

3

-58

16

Denmark

2

-12

17

Scotland

2

-16

18

Finland

2

-56

19

Norway

2

-94

20

Belarus

1

-38

21

Austria

1

-52

22

Northern Ireland

1

-144

23

Russia

0

-140

24

Greece

0

-180

Overall good start to the ETC for team Ireland with one win, one draw, and one loss. Team NI didn’t get such a good start, with one draw and two losses. Here’s the matchups for Round 4 tomorrow morning:

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My posting schedule has been pretty haphazard so far. So I’m turning to you to figure out what update schedule works best. I have a poll set up here so please do vote your preference, also feel free to leave a comment (either here or on the facebook page) on what kind of update schedule is best, and what topics you’d like to see me cover.

Some requests in the pipeline are:

Assault cannon vs Lascannon for penetrating AV14

Blast weapons

Other dice combinations such as 3d6 drop the highest, 4d6, 2d6 reroll

Your wish is my command!

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Deep striking is a high risk/reward technique that can get your units anywhere on the table, in one fell swoop. But when things go wrong, they can go very wrong, and on more than one occasion I’ve lost a 225 point unit of Obliterators to a bad scatter. For that reason I often take some 3 man chaos terminator squads so I only risk 105 points for a chance at a cheeky melta shot. But how should I be placing them when I deepstrike? Consider the following:

Chaos Lord Harleck Wynne faces a wall of Imperial Guard tanks. He has to deepstrike his terminators, as any walking squad or vehicle will be wiped out as it approaches. Where should his combi-melta armed Chaos Terminators be placed to minimise the risk of mishap? Where should they be placed to maximise the chances of getting into melta range? Where should they be placed to get a balanced risk of mishap versus melta range?

Ok, so deep striking is governed by scatter dice. It’s a 6 sided die with two ‘HIT’ faces, and four faces with an arrow. Place your model where you want him, and roll. If you get a HIT then you land on target, if you get an arrow, then you scatter 2d6 inches away in the direction indicated by the arrow. Because the distance is governed by 2d6, the distance follows a pattern already described here such that results of 7″ are the most likely and 2″ and 12″ are the least likely.

The arrows complicate matters as they don’t comply with the discrete probability that I normally use for these calculations, but we’ll touch on that later.

So, with two HIT faces out of 6, we have a 33% chance of landing on target, and a 67% chance of scattering. If we ignore direction for a moment, then we can take a look at the odds of how far you’ll deviate from your intended location:
That was pretty much as far as my analysis went until quite recently. This approach clouded my thinking, as I saw it as a straight up question of distance, so I may as well get super close to the enemy as the ‘most likely’ scatter distance was 7″. Case closed, right?

Wrong!

If you don’t get a HIT, then it’s all about the arrows. Let’s imagine a model with a 25mm base put on the table in his desired deepstrike position. He can scatter up to 12″ in any direction, so lets consider a 25″ wide circle as the total space we could end up in (e.g. up to 12″ to left + 1″base + 12″ to the right gives us the 25″, see below).

Time for a fancy graph. So I plot an area of 25″ by 25″, and represent the probability of landing at a particular point as a height, so we we get a sort of mountainous terrain where the highpoints are where you are likely to land, and the lowpoints are where you are unlikely to land. In the first instance lets look at the widest case. So you have a 33.3% chance of landing on target (i.e. a HIT), and a 66.6% chance of scattering. See below:

As you can see in terms of a single point, the target at the centre is far and away the single most likely final destination. In fact the difference is so extreme that all you can see of the scatter is some light ‘fuzz’ in a ring around the centre. So the first point to note is that if you do scatter it would appear that you could end up pretty much anywhere in that 25″ circle we described earlier. But that’s not particularly enlightening, so lets take a closer look at the ‘fuzz’.

I now remove the HIT from the chart, and the scale can then be changed to show the variation in odds for the scatter results. It’s worth noting that I didn’t solve this analytically so we don’t get a smooth and pretty set of results, we get a somewhat noisy set of peaks and valleys. But it’s still good enough to gain some insights and is still essentially representative of how it works in reality.

So as you can see from the dark blue peaks, the most likely area to scatter into is a ring around the target point, (specifically a ring with its edges about 5″ to 9″ away from the target point). This is an expected result from our knowledge that the scatter follows the same triangular shape of the old 2d6 chart. Do note how low the odds of landing at any particular point is: about 0.2% to 0.4%, tiny! Working through the numbers, here’s a simplified version:

So this is a lot of exposition and I haven’t addressed the opening question at all! What about those terminators?

Based on the calculations above, I carved out the probability of landing in a ‘safe’ area depending on how far away you place the terminators. But that in isolation is not enough. We want the terminators to land within 6″ of the tanks to get some hot melta goodness going. So here I’ve plotted the odds of landing safely for a given distance, and also the odds of ending up safe AND within melta range for a given drop point (i.e. the point you selected to drop at, not where you end up after scattering). So on this graph the x-axis is the distance from the tank wall you place the model initially, (i.e. before rolling for scatter).

The results weren’t quite what I was expecting going in, though do bear in mind that these findings are only true for the specific set up of the question – this graph isn’t a general rule for all deep strike situations!

So, what does this show? Well, assuming the parking lot of tanks is the only other unit in the area then unsurprisingly the further away you place them the less likely they are to scatter on to the enemy and mishap. But playing it safe won’t necessarily get you within the all important 6″ melta range. Here’s the interesting bit, I had originally thought that putting the terminators 1″ away from the tanks would get you the highest probability of being in melta range with a trade off of slightly higher odds of mishap. But I was quite wrong. The odds of getting safely in melta range stay pretty flat if you originally place the model between 1″ and 6″ away, but the odds of a mishap are about 45% at 1″ but fall to about 25% at 6″. So the tradeoff I mentioned in my opening question, doesn’t really exist – you can play it (relatively) safe and still go for the close range shot.

Lesson learned, drop those terminators about 5 or 6 inches away and you’re playing the right odds.

So how about a more general rule of thumb then? This specific case aside, how do we make better deepstriking decisions on the fly? In my opinion, the best general approach is to think in terms of area. Visualise the 25″ circle around any particular drop point (some assistance here and here), and then look at the friendly and enemy units in that circle. Now imagine a 1″ buffer around enemy units, and try to estimate what fraction of the circle’s area is covered by all the units and that buffer. This is key to estimating the risk.

I’ve illustrated a few simple examples below; in each case the centre of the circle is where you initially place the model (i.e. before rolling for scatter), and the red areas have units or other features (such as impassable terrain) that would cause a mishap (don’t forget the 1″ buffer around enemy units!). Do note I’m assuming that the centre point is a legal placement. Also note the maths below isn’t quite exact, but is good enough for tabletop guesstimation.

So there you have it – even deep striking right up into someone’s face is not quite as risky as it looks.

As a change of pace, I’m foregoing probability for the sake of a proper hobby update. My main army is Alpha Legion (using codex chaos space marines) but I really miss having the use of cultists as human operatives. I’ve had a hankering to run a renegade guard list with Alpha Legion operatives for a while now, and with that in mind I’ve been slowly amassing some forgeworld renegades over the last 12 months.

The forgeworld models are fantastic and here’s what I’ve picked up so far.
9 enforcers

50 bods and 3 weapon teams
command squad

6 tank crew

7 psykers

Obviously I’ll need to get a lot of vehicles to make a viable guard army, but I’m looking at using some fairly heavy conversion to set the army apart from a ‘normal’ Imperial Guard army. More updates soon, and there are pictures available on the accompanying FaceBook page.

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The previous post on the probabilities for making lots of saves generated a bit of interest, and (as usual) some clever readers pointed out scenarios that should bear further analysis. Altmann from the Penny Arcade forums asked:

“Can you work in the probability of making 4+ feel no pains as well? I know we’re getting into NASA shit but I’m curious”

Followed by Joe “Maynard” Cullen (of WarHeads fame) who pointed out that some wargear items also add complexities:

So in a similar fashion to my ultimate Ballistic Skill chart, I took it upon myself to rank the performance of a variety of armour types with rerolls, with Feel No Pain (FNP), and just plain regular saves. This will give some insight into the relative merit of the saving throws we normally encounter in 40k.

As with Stacking the Odds Part I, the chart shows how likely each type of save is to take no casualties from an increasing number of saves.

So chart number one:
This charts the various types of save (and combinations) showing the odds of taking no casualties for up to 6 saves (I cut it off at 6 as about half of them approach zero at this point). The legend on the right shows the ranking from best to worst with a 2+ rerollable save being the best, and a regular 6+ save being the worst. The sharper eyed in the audience may notice that some of the save types listed in the legend don’t show up in the graph – namely “5+ FNP” and “3+ FNP”. Rest assured this isn’t an error, it is simply that they are coincidentally covered by other save types that perform identically. So a 5+ with Feel No Pain save works out the same as a regular 3+ save, and a 3+ with Feel No Pain save works out the same as a regular 2+ save.

Or do they?

The calculations are correct, but you need to interpret the data in the context if the game itself. So saving on a 5+ followed by a 4+ for FNP is statistically the same as a 3+, until you get hit by an AP5 or AP4 weapon, at which point all you get is the FNP, which is just a 4+ save (as you can see from the chart is a lot worse than a 3+). The FNP could also be blocked by a high strength AP- weapon, leaving you with just a 5+.

In a similar vein, the 3+ with FNP is the same as a 2+, but what if they got hit by a battle cannon? the 3+ is negated by AP3, and (assuming we’re talking about T4 units) the FNP is negated by the instant death rule. So no saves of any kind! But a squad of terminators would still get their 2+ and (assuming that 5 are wounded by the blast) they have a 40% chance of taking no casualties at all!

So what about those opening questions? Well Altmann was interested in the effect of FNP on terminators, and to show the difference I’ve scaled the number of saves taken up to 30, and dropped the weaker save types.

The effect is actually pretty strong, if we take say 20 saves, a regular terminator squad has only a 3% chance of being unharmed while the FNP terminators have an 18% chance (again assuming that they aren’t hit by something that negates FNP!)

Joe’s suggestion of the Wolf Tail Talisman (WTT) is charted below. Assuming the squad has power armour, then it works out quite close to (but slightly better than) a 4+ reroll, and worse than a 2+ save.

This should let you compare the various save types available to you, but don’t forget the context of how saves and FNP get negated! If there are any other save types you want to see included, then please do leave a comment.

This is a topic I touched on briefly before, but I think it’s time for a more comprehensive review. As I’ve said previously, many players have bad instincts for how odds ‘stack up’, and I often hear comments like ‘if a terminator has to make 6 saves, you’d expect one to fail’. Time to challenge some assumptions.

A squad of terminators come under bolter fire and have to take 6 saves, what is the probability that they take no casualties? A squad of marines come under a similar hail of bolter fire and also have to take 6 saves, what is the probability that they take no casualties?

At tournaments, and during club play, you may often hear cries of consternation as someone can’t believe that their opponent just made X number of saves in a row. I think this ties in to certain types of (erroneous) expectations. Nothing in this game of dice is certain; and you can never get a 100% guarantee of success.

That sounds pretty trite, and taken purely at face value, it is. But I’m trying to get at something a little deeper. If a terminator has to make ten saves, or a thousand saves, there is always a chancing of making it, but the odds don’t scale in the way that people expect. Simply put: a thousand bolter shots is not one thousand times more likely to kill a terminator than one bolter shot; nor is ten bolter shots ten times more likely. As the number of saves to be taken increases, the odds of survival go down in an exponential way (for the non math types that means they start high, and gradually get lower but never quite reach zero).

Here’s one I prepared earlier:

This graph shows how terminator armour and power armour behave as the number of saves to be taken goes up. Terminator saves are in red, and marine saves are in blue. The x-axis represents the number of saves that have to be made, and the y-axis shows the odds of the squad taking no casualties for the corresponding number of saves.

So this gives us the answer to the opening question. If the termies have to take 6 saves, then there’s a 33% chance of them taking no casualties (i.e. find 6 on the x-axis and then look at the corresponding point on the y-axis for the red line). The other group aren’t so hot, if the marines have to take 6 saves, then there’s a 9% chance of them taking no casualties.

If we extend the analysis a bit, the marines have only a 1% chance of taking no casualties from 12 saves, whereas the terminators have a more respectable 11% chance. Even at 18 saves the terminators still have a 4% chance of walking away without a scratch. Now 4% may sound like very long odds, but in truth its not far off the odds of getting a ‘perils of the warp’ result for a psyker. So not something you’d see a lot, but hardly beyond belief.

One (slightly esoteric) point to note is that this is a ‘memoryless’ system. This gets a bit subtle, but what I mean is that the current odds aren’t affected by what happened before. So if a squad of terminators all survived 6 saves last turn and are now facing 6 more saves in this turn, the odds don’t stack to 11% (i.e. for 12 saves), they stay at 33% (for 6 saves). Whatever happened in the past doesn’t affect your current action.

Now that we’ve covered some specifics, I’ve taken the liberty to graph the behaviour of saves from 2+ to 6+ when having to make up to 6 saves in one block. Each coloured line represents a corresponding type of save from red for a ‘terminator’ save, blue for power armour, and so on through to the grey line for a 6+ save.

This follows the same pattern as the previous, it just shows more armour types. But as an interesting illustration, based on the graph check out the following:

A terminator squad taking 6 armoursaves has a 33% probability of taking no casualties

A terminator squad taking 6 storm shield saves has a 9% probability of taking no casualties

A terminator squad taking 6 cover saves (4+) has a 2% probability of taking no casualties

A terminator squad taking 6 invulnerable saves (5+) has a 0.14% probability of taking no casualties

Do keep this in mind the next time you fire your hydras at my obliterators!

Like this:

As a quick break from the usual TheoryHammer, and just to show it’s not all about statistics, I’ve added some images to the resources section for those of you who are into terrain building. It’s a set of miniature posters for the 40k/necromunda setting.

Gabriel_Pitt at the excellent Penny Arcade forums provided the source material – though he did mention that they apparently come from GW site (not that I can find them there now).

Here’s an example of how they were used in one of his terrain projects: